Developing Multiplicative Thinking

The Power of

Grouping in K-2

Candy Standley, Math Specialist

Kristen Brink, Math Specialist

Culpeper County Public Schools

September 26, 2014

What is Multiplicative Thinking?

A capacity to work flexibly and efficiently with the

concepts, strategies, and representations for

multiplication.

As students develop multiplicative thinking, focus is

on

◦ groups of equal size

◦ the number of groups, and

◦ the total amount.

Multiplicative thinking leads to an understanding of

fractions, decimals, percents, ratio, and algebra.

Different Ways to Count

More and more, we

saw students

counting sets without

understanding that

ones, twos, fives,

and tens represent

groups.

Red Flags at STRIDES

Student brought to STRIDES team

because she couldn’t recognize and name

numbers to 10.

Couldn’t count a set up to 10.

Couldn’t tell more or less within 10.

Parent commented that student

successfully counts to 100 by fives and

tens everyday.

Prompted case study in kindergarten.

Kindergarten SOL K.4

The student will…

a) count forward to 100 and backward

from 10;

b) identify one more than a number

and one less than a number; and

c) count by fives and tens to 100.

Understanding the Standard

The patterns developed as a result of skip

counting are precursors for recognizing

numeric patterns, functional relationships,

and concepts underlying money, time

telling, and multiplication. Powerful models

for developing these concepts include, but

are not limited to, counters, hundred

chart, and calculators.

Understanding the Standard

continued… Skip counting by fives lays the foundation

for reading a clock effectively and telling

time to the nearest five minutes, counting

money, and developing the multiplication

facts for five.

Skip counting by tens is a precursor for

use of place value, addition, counting

money, and multiplying by multiples of 10.

3rd Grade Needs Assessment

Solve practical problems that apply knowledge of multiplication.

Tell time to the nearest minute.

Make change.

Elapsed time.

Read a thermometer.

Solve a problem using a pictograph.

Analyze and interpret information on a graph.

Remarks by President Clinton

on the Education Standards 1997

“In no other country in the world did

performance in math drop from above

average in 4th grade to below average in 8th

grade…We are doing a very good job in the

early grades, but we’ve got a lot more work

to do in the later ones.”

Remarks by President Clinton on Education Standards, The White House,

June 10, 1997

Pause to Ponder…

Where does the

mathematical decline

really begin?

Clark and Kamii, 1996

Research has shown that multiplicative

thinking develops slowly in children, over

long periods of time.

Students need practice with tasks that

help develop multiplicative thinking.

The Value of Experience

While the everyday experience of most 5

year-olds supports intuitive ideas of

getting/having more (addition),

losing/having something less (subtraction),

fair shares/sharing (division), and doubling,

it generally does not support a notion of

counting in equal groups or repeated

addition. (Siemon, Breed, Virgona)

Counting Collections

https://www.teachingchannel.org/videos/s

kip-counting-with-kindergarteners

How might this type of activity develop

multiplicative thinking in 5, 6, and 7 year

olds?

In what other grades might this activity be

appropriate?

What mathematical concepts might this

type of activity support?

Butterfly House

Butterfly HouseHow many complete butterflies can be

made with:

29 wings

8 bodies

13 feelers

Consider your mental math while you

attempt to solve this problem….where is

your multiplicative thinking?

Butterfly House

What type of thinking or skills do you

need in order to solve problems like this

efficiently?

Where is the multiplicative thinking?

Where does the thinking fall apart for

students who haven’t transitioned to

multiplicative thinking?

(Siemon)

Feeding Fish

Folding Paper

If you fold an 8 ½ x 11 sheet

of paper in half 6 times, how

many sections would you

create?

References

From Additive to Multiplicative Thinking—The Big Challenge of the

Middle Years

Dianne Siemon, Margarita Breed & Jo Virgona

The Power of Paper-Folding Tasks: Supporting Multiplicative Thinking and

Rich Mathematical Discussion

Erin E. Turner, Debra L. Junk, Susan B. Empson

Educating Teachers to Teach Multiplicative Structures in the Middle

Grades

Judith Sowder, Barbara Armstrong, Susan Lamon, Martin Simon,

Larry Sowder, Alba Thompson

Identification of Multiplicative Thinking in Children in Grades 1-5

Faye B. Clark, Constance Kamii